but Guy Terjanian found the following 2-adic counterexample for d = 4. Define Then G has the property that it is 1 mod 4 if some x is odd, and 0 mod 16 otherwise. It follows easily from this that the homogeneous form G(x) + G(y) + G(z) + 4G(u) + 4G(v) + 4G(w) of degree d = 4 in 18 > d2 variables has no non-trivial zeros over the 2-adic integers. Later Terjanian showed that for each prime p and multiple d > 2 of p(p − 1), there is a form over the p-adic numbers of degree d with more than d2 variables but no nontrivial zeros. In other words, for all d > 2, Yd contains all primes p such that p(p − 1) divides d. gave an explicit but very large bound for the exceptional set of primes p. If the degree d is 1, 2, or 3 the exceptional set is empty. showed that if d = 5 the exceptional set is bounded by 13, and showed that for d = 7 the exceptional set is bounded by 883 and for d = 11 it is bounded by 8053. See also Brauer's theorem on forms Quasi-algebraic closure Notes References (Corollary 5.4.19) Model theory Theorems in number theory